Eigenvector methods are gaining increasing acceptance in the area of spectrum estimation. This paper presents a successful attempt at testing and evaluating the performance of two of the most popular types of subspace techniques in determining the parameters of multiexponential signals with real decay constants buried in noise. In particular, MUSIC (Multiple Signal Classification) and minimum-norm techniques are examined. It is shown that these methods perform almost equally well on multiexponential signals with MUSIC displaying better defined peaks

Jibia, Abdussamad U.

Salami, Momoh-Jimoh E.

English

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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/216573576Performance Evaluation of Music and Minimum Norm EigenvectorAlgorithms in Resolving Noisy Multiexponential SignalsArticle · January 2007CITATIONS8READS452 authors:Some of the authors of this publication are also working on these related projects:Development of ANN based technique for TVAR parameter estimation View projectDevelopment of an intelligent scorpion detection technique using vibration analysis View projectAbdussamad JibiaBayero University, Kano17 PUBLICATIONS   44 CITATIONS   SEE PROFILEMomoh SalamiInternational Islamic University Malaysia140 PUBLICATIONS   933 CITATIONS   SEE PROFILEAll content following this page was uploaded by Abdussamad Jibia on 17 May 2014.The user has requested enhancement of the downloaded file.  Abstract—Eigenvector methods are gaining increasing acceptance in the area of spectrum estimation. This paper presents a successful attempt at testing and evaluating the performance of two of the most popular types of subspace techniques in determining the parameters of multiexponential signals with real decay constants buried in noise. In particular, MUSIC (Multiple Signal Classification) and minimum-norm techniques are examined. It is shown that these methods perform almost equally well on multiexponential signals with MUSIC displaying better defined peaks.  Keywords—Eigenvector, minimum norm, multiexponential, subspace.  I. INTRODUCTION HE task of analyzing multiexponential signals with real decay constants has been the subject of many research efforts for more than five decades. This is because, straightforward as their analysis may appear, such signals do not form an orthogonal base. The signals are usually of the form: 1( ) ( )kMkkS A e nλ ττ τ−== +∑                         (1)                                                                                                                             The unknown parameters are usually the number of components (M), the amplitudes (Ak) and the decay rates (λk). n(τ) represents the noise, considered to be white. The facts that these signals occur in many natural and artificial phenomena makes their analysis even important. Some of the areas they occur include: temperature modulation of Metal-oxide semiconductor (MOS) sensors [6], fluorescence decay analysis [8], Electromagnetic field analysis [7], nuclear magnetic resonance [3], transient spectroscopy [5], compartment analysis in physiology [4], etc. Several methods have been documented for the analysis of  Manuscript received September 10, 2007.  M. J. E. Salami is a professor of Digital Signal Processing and Deputy Dean (Postgraduate and Research) of the Kulliyyah of Engineering, International Islamic University, Malaysia (e-mail: momoh@iiu.edu.my). A. U. Jibia is a staff of Bayero University, Kano, Nigeria and a PhD student in the Department of Electrical and Computer Engineering, Kulliyyah of Engineering, International Islamic University, Malaysia (phone: +60196184521;  e-mail: ajumar2000@yahoo.com)  this class of signals. These methods range from theoretical exercises [9] to more practical efforts [1], [2], and [8]. In this paper, an attempt has been made to compare the performances of two eigenvector methods in the analysis of multiexponential signals buried in noise. Subspace methods were primarily developed for the analysis of complex exponentials buried in noise and do not therefore fit to our analysis in the first place. To make it suitable for analysis by eigenvector methods, multiexponential signal has been subjected to a preprocessing procedure that involves Gardeners' transformation followed by a deconvolution stage. The Gardeners' transformation transforms the signal into a convolution model which when deconvolved results in a sum of complex exponentials in noise assumed to be white.  The subspace methods are then applied to the deconvolved data in turn. The single most important property of eigenvector (also called subspace) methods is that they produce unbiased estimates with infinite resolution regardless of signal-to-noise ratios [14]. The methods considered in this paper are the Multiple Signal Classification (MUSIC) and the minimum-norm methods. II. SIGNAL PREPROCESSING The main objective of the signal preprocessing is to obtain an orthogonal representation of exponential data by applying Gardners' transformation to (1) followed by Fourier processing.  Initially, (1) is expressed as   1( ) ( ) ( ) , 0Mk kkS A p nτ λτ τ τ== + < <∞∑                       (2)                  where the basis function p(τ) = exp(-τ). This equation can be rewritten as 0( ) ( ) ( ) ( )S g p d nτ λ λτ λ τ∞= +∫                     (3)                   where 1( ) ( )Mk kkg Aλ δ λ λ== −∑                            (4)                   and it contains all the parameters to be determined. Multiplying both sides of (3) by τα and applying the Abdussamad U. Jibia, and Momoh-Jimoh E. Salami Performance Evaluation of Music and Minimum Norm Eigenvector Algorithms in Resolving Noisy Multiexponential Signals T World Academy of Science, Engineering and Technology 32 200724  Gardner transformation, τ = et and    λ = e-r results in a convolution integral  ( ) ( ) ( ) ( )y t x h t d v tλ λ λ∞−∞= − +∫                    (5)                                                                                                                             where  )()exp()(),()exp()()},(})1exp{()()},{exp()exp()(tttenttvandeptthegttxtSttyαααα==−==− Taking the Fourier transform of (5) and performing inverse filtering followed by inverse Fourier transformation yields  1( ) ( ln )Mk kkx t B tδ λ== +∑                            (6)                                                                                                                             where αλ −= )( kkk AB . Some of the drawbacks of this method have been highlighted in [18].  A discrete form of (5) is obtained by sampling y(t) at a rate of  1/Δt Hz, yielding the discrete convolution maxmin[ ] [ ] [ ] [ ]nm ny n x m h n m v n=−= − +∑                             (7)                                                                                                                              where N = nmax-nmin+1, nmax and nmin represent respectively the upper and lower data cut-off points. The criteria for the selection of these sampling conditions have been thoroughly discussed in [2] and [4]. III. GENERATION OF THE DECONVOLVED DATA Taking the DFT of (7) yields  ( ) ( ) ( ) ( )Y k X k H k V k= +   from which the deconvolved data can be generated according to  ( ) ( )ˆ ( ) ( ) ,( ) ( )Y k V kX k X kH k H k= = +                      (8)                                                                                                                                                              for 0 ≤ k ≤ N-1, where Y(k), X(k), H(k), and V(k) represent respectively the DFT of y(n), x(n), h(n), and v(n). This inverse filtering operation yields deconvolved data with decreasing SNR for increasing values of k. To alleviate this problem an optimal inverse filtering procedure is used for generating X(k). In this approach H (k) is modified so that the deconvolved data is generated according to [15] as  ( ) ( )ˆ ( ) ,2[ ( ) ]Y k H kX kH k μ∗=+                             (9)                                                                                                                                                    where the symbol * denotes complex conjugate. For high SNR, μ should be small and of the same order of magnitude as the attenuation of H (k) at the cut-off frequency. However, as the SNR of the data decreases the choice of the optimum value of μ in (10) is best determined by experimental testing [3]. Denoting the truncated data as f (k) and based on (6), (8) and (9), then we have ( ) exp{ ln } ( ),1Mf k A jk ki iiω λ ε= Δ +∑=                                   (10)                  For k = 1, 2, ……., 2N0 +1; N0 ≤ (N/2)-1 is the truncation point and ε(k) is the deconvolved noise. IV. SUBSPACE SIGNAL PROCESSING Eigenvector (also called subspace) methods are based on a certain time-window of length P over which the signal model is characterized in form of a vector. Consider the signal f(k) from equation (10) at its current and future P- 1 values. The time-window can be written as  f(k)= [f(k)  f(k+1)……f(k+ P-1)]                       (11)                                                                                                                      We can then write f(k) in terms of length-P time-window as   ∑=Δ +=+=Mijkii kwkskwerAkf i1ln )()()()(ln)( λωλ           (12)                   where w(k)= [w(k) w(k+1) ….. w(k+P-1)]T is the time-window vector of white noise and  r(lnλ)= [1 ejΔωlnλ ……. ejΔωlnλ(P-1)]T is the time-window signal vector at each particular value of lnλ. The autocorrelation matrix of f(k) may be expressed in terms of its eigendecomposition as  ∑=Λ==MiHHiiiff VVvvR1λ                       (13)                                                                                 where Λ is a diagonal matrix of the eigenvalues in descending order on the diagonal (λ1≥λ2≥ ……….≥λM) while the columns of V are the corresponding eigenvectors. The M largest eigenvalues correspond to the signal and the remaining eigenvalues have equal values and correspond to the noise. Rff can thus be partitioned into two portions, one due to the signal and the other due to the noise eigenvectors  Hwwwsssff VVVVR2σ+Λ=                           (14)                  where  Vs= [v1  v2 …….vM] and Vw= [vM+1 ……. vP]          (15)  are matrices whose columns consist of signal and noise eigenvectors respectively. Λ is an MxM diagonal matrix containing the signal eigenvalues. Thus the P-dimensional World Academy of Science, Engineering and Technology 32 200725  subspace has been broken into two subspaces, i.e. the so-called signal and noise subspaces. The matrices that project an arbitrary vector on the signal and noise subspaces are [12]: Hsss VVP =   and  Hwww VVP =                       (16)                                                                                                                             Since the signal and noise subspaces are orthogonal,  0=swVP    and   0=wsVP                           (17)                                                                                                                             all the lnλ vectors from equation (12) must lie completely in the signal subspace. This means )(ln)(ln iis rrP λλ = ; 0)(ln =iwrP λ                 (18)                                                                                                                The above concepts are central to the two subspace methods presented below. A. MUSIC Algorithm The MUSIC algorithm was proposed [10] as an improvement on the Pisaranko harmonic decomposition (phd) which limited the length of P to P= M+1. In the MUSIC method, the time window is allowed to be P> M+1, thus giving a subspace of dimension greater than 1. For each eigenvector vp in the noise subspace (M<p≤P),  0)()(ln1)1)(ln( == ∑=−Δ−PkkjmpmH mekvvr λωλ          (19)                                                                         for all the M values of lnλ. Thus, if we compute the pseudospectrum of each noise eigenvector as  2ln2)ln)(1)(ln1(λωλωλ ΔΔ ==jppHjPeVvreR    (20)                                                                  the polynomial )( ln λωΔjp eV has P-1 roots, M of which corresponds to the lnλ of the complex exponentials. These roots produce M peaks in the pseudospectrum. The pseudospectra of all P-M noise eigenvectors share these roots that are due to the signal subspace. The remaining roots of the noise eigenvectors occur at different frequencies and may produce extra peaks in the pseudospectrum if close to the unit circle. The solution to this is to average out the P-M pseudospectra of the individual noise eigenvectors:   ∑∑+=Δ+=Δ == PMpjpPMppHjmusiceVvreR12ln12)ln)(1)(ln1(λωλωλ        (21)                                                                                                This is known as the MUSIC pseudospectrum [12].  B.  Minimum Norm Algorithm Like the MUSIC method, the time-window P is allowed to be P> M+1. The minimum-norm method uses an arbitrary vector  b= [b(1)  b(2) ……… b(P)]T                     (22)                                                                                                                    constrained to lie on the noise subspace. For any arbitrary vector that lies in the noise subspace  Pwb= b           and  Psb= 0                       (23)                                                                                                                     where 0 is the length-M zero vector. The minimum norm seeks to minimize the norm of b in order to avoid spurious peaks in the pseudospectrum. From equation (16), the norm of a vector b contained in the noise subspace is  bPbbbb wHH ==2                          (24)                  Since an unconstrained minimization of this norm will produce the zero vector, the first element of b is constrained to be unity, i.e. 11 =bHδ                                         (25)                   The solution to this can be found by using Lagrange multipliers [12] as 111δδδwHwmn PPb =                            (26)                   The lnλ estimates are then obtained from the peaks in the pseudospectrum of the minimum norm vector. V. SIMULATION RESULTS A MATLAB programme was written and run for the above algorithms and the performance of the above subspace techniques in analyzing multiexponential signals examined using the signals S1(τ) and S2(τ), where  )(10525.0)( 10525.01 ττ τττττ neeeeeS +++++= −−−−−     (27)                   )(5.0)( 5.02 ττ ττ neeS ++= −−                        (28)                                                                                      The pseudospectrum of S1(τ) and S2(τ) are shown in Fig. 1 through Fig. 4 for the three techniques and Tables I and II give the estimate for lnλi for S1(τ) and S2(τ) respectively. The results show that the MUSIC algorithm produces more defined peaks. For the two algorithms, it is evident that the estimates do not vary significantly with SNR and they are accurate within the tolerance of about ±0.02. We may therefore conclude that for multiexponential signals in noise, MUSIC is the better algorithm even though the minimum norm performs almost equally well.   World Academy of Science, Engineering and Technology 32 200726  -4 -3 -2 -1 0 1 2 3 4-120-100-80-60-40-20020Magnitude (dB)Log (lambda)  Fig. 1  MUSIC pseudospectrum of S1(τ) with SNR of 40dB -4 -3 -2 -1 0 1 2 3 4-120-100-80-60-40-20020Magnitude (dB)Log (lambda)  Fig. 2 MUSIC pseudospectrum for S2(τ) at SNR of 40dB -4 -3 -2 -1 0 1 2 3 414151617181920Magnitude (dB)Log (lambda)  Fig. 3 Min-Norm pseudospectrum of S1(τ) with SNR of 40dB -4 -3 -2 -1 0 1 2 3 411121314151617181920Magnitude (dB)Log (lambda)  Fig. 4 Min-norm pseudospectrum for S2(τ) at SNR of 40dB    TABLE I    VALUES OF lnλi FOR DIFFERENT SNR FOR S1(τ)   TABLE II    VALUES OF lnλi FOR DIFFERENT SNR FOR S2(τ)  VI. CONCLUSION The performance of two eigenvector methods has been examined in the analysis of multiexponential signals in noise. It has been shown that both MUSIC and Minimum norm subspace techniques perform equally well with MUSIC resulting in better defined peaks at the lnλ values. It has also been shown that the performance of the two methods does not deteriorate with decreasing SNR. REFERENCES   [1] D.G. Gardner, J.C. Gardner, G. Lush, and W.R. Ware, “Method for the analysis of multicomponent exponential decay curves”, Journ. Chem. Phys., vol. 31, 1959, pp. 978-986. [2] M.J.E. Salami, “ARMA models in multicomponent signal analysis”, Ph.D. Dissertation, Dept. of Electr. Eng., University of Calgary, Calgary, Canada. 1985. [3] M.J.E. Salami, and S.N. Sidek, “Performance evaluation of the deconvolution techniques used in analyzing multicomponent transient signals”, in Proc. IEEE Region 10 International Conference on Intelligent System and Technologies for the New Millennium (TENCON 2000), Vol. 1, 2000, Kuala Lumpur, pp. 487-492 [4] M.J.E. Salami, and S.N. Sidek, “Parameter estimation of multicomponent transient signals using deconvolution and ARMA modeling techniques”, Journal of Mechanical systems and signal processing, Vol. 17, Issue 6, pp. 1201 – 1218, Academic Press, UK. [5] M.J.E. Salami, “High-resolution decay rate estimation in multiexponential signal analysis”, Pro. 3rd Intern. Conf. on Acoustical and vibratory surveillance methods and diagnostic techniques, 1998, pp. 379-388. [6] R. Gutierrez-Osuna, A. Gutierrez-Galvez, and N. Powar,. "Transient response analysis for temperature-modulated chemoresistors", Sensors and Actuators Journal, pp. 57-66, 2003. [7] T. K. Sarkar, and O. Pereira, "Using the matrix pencil method to estimate the parameters of a sum of complex exponentials" IEEE Antennas and Propagation Magazine, vol. 37. No.1, 1995, pp. 48-55. [8] D. F. Eaton, "Recommended methods for fluorescence decay analysis", Journal of Pure and Applied Chemistry, vol. 62, No 8, 1990, pp. 1631-1648. World Academy of Science, Engineering and Technology 32 200727  [9] D. Lawunmi, "A theoretical analysis of exponentially decaying time series" Journal of Measurement Science and Technology, IOP Publishing Ltd., UK. 1997, pp. 703-706. [10] G. Bienvenu, and L. Kopp, "Optimality of high resolution array processing using the eigensystem approach", IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 31(10), pp. 1235-1248, 1983. [11] M.J.E. Salami and Z. Ismail, "Analysis of multiexponential transient signals using interpolation-based deconvolution and parametric modeling techniques", in Proc. IEEE International conference on Industrial Technology, vol. 1, 2003, pp. 271-276. [12] D.G. Manolakis, V.K. Ingle, and Kogon, S.M., Statistical and Adaptive Signal Processing, Artech House, Inc., Norwood, 2005. [13] M.H. Hayes, Statistical Signal Processing and Modeling, John Wiley and sons, Inc., New York, 1996. [14] S. J. Orfanidis, Optimal Signal Processing. McGraw-Hill Publishing Company, 2007. [15] Salami M.J.E., “High-resolution decay rate estimation in multiexponential signal analysis”, in Pro. 3rd Intern. Conf. on Acoustical and vibratory surveillance methods and diagnostic techniques, 1998, pp. 379-388.  [16] S.M. Kay, and S.L. Marple, "Spectrum   analysis   –   A modern perspective", in Proc. of the IEEE. 69(11), 1981, pp. 1380-1419.    [17] M.J.E. Salami, S.T Nichols, and M.R. Smith, “A SVD-based transient error method for analyzing noisy multicomponent exponential signals”, in Proc. ICASSP87, 1987, pp. 677- 680. [18] M.R. Smith, S. Cohn-Sfetcu, and Buckmaster H.A., “Decomposition of multicomponent exponential decays by spectral analytic techniques”, Technometrics, vol. 18, pp. 467-482, 1976. World Academy of Science, Engineering and Technology 32 200728View publication stats

Performance evaluation of music and minimum norm eigenvector algorithms in resolving noisy multiexponential signals

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Performance evaluation of music and minimum norm eigenvector algorithms in resolving noisy multiexponential signals

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